Adaptive fault-tolerant routing in hypercube multicomputers

A connected hypercube with faulty links and/or nodes is called an injured hypercube. To enable any non-faulty node to communicate with any other non-faulty node in an injured hypercube, the information on component failures has to be made available to non-faulty nodes so as to route messages around the faulty components. A distributed adaptive fault tolerant routing scheme is proposed for an injured hypercube in which each node is required to know only the condition of its own links. Despite its simplicity, this scheme is shown to be capable of routing messages successfully in an injured hypercube as long as the number of faulty components is less than n. Moreover, it is proved that this scheme routes messages via shortest paths with a rather high probabiltiy and the expected length of a resulting path is very close to that of a shortest path. Since the assumption that the number of faulty components is less than n in an n-dimensional hypercube might limit the usefulness of the above scheme, a routing scheme is introduced based on depth-first search which works in the presence of an arbitrary number of faulty components. Due to the insufficient information on faulty components, the paths chosen by the above scheme may not always be the shortest. To guarantee that all messages be routed via shortest paths, it is proposed that every mode be equipped with more information than that on its own links. The effects of this additional information on routing efficiency are analyzed, and the additional information to be kept at each node for the shortest path routing is determined. Several examples and remarks are also given to illustrate the results

A class of hierarchical graphs as topologies for interconnection networks

We study some topological and algorithmic properties of a recently defined hierarchical interconnection network, the hierarchical crossed cube HCC(k,n), which draws upon constructions used within the well-known hypercube and also the crossed cube. In particular, we study: the construction of shortest paths between arbitrary vertices in HCC(k,n); the connectivity of HCC(k,n); and one-to-all broadcasts in parallel machines whose underlying topology is HCC(k,n) (with both one-port and multi-port store-and-forward models of communication). Moreover, (some of) our proofs are applicable not just to hierarchical crossed cubes but to hierarchical interconnection networks formed by replacing crossed cubes with other families of interconnection networks. As such, we provide a generic construction with accompanying generic results relating to some topological and algorithmic properties of a wide range of hierarchical interconnection networks

Reliability Analysis of the Hypercube Architecture.

This dissertation presents improved techniques for analyzing network-connected (NCF), 2-connected (2CF), task-based (TBF), and subcube (SF) functionality measures in a hypercube multiprocessor with faulty processing elements (PE) and/or communication elements (CE). These measures help study system-level fault tolerance issues and relate to various application modes in the hypercube. Solutions discussed in the text fall into probabilistic and deterministic models. The probabilistic measure assumes a stochastic graph of the hypercube where PE\u27s and/or CE\u27s may fail with certain probabilities, while the deterministic model considers that some system components are already failed and aims to determine the system functionality. For probabilistic model, MIL-HDBK-217F is used to predict PE and CE failure rates for an Intel iPSC system. First, a technique called CAREL is presented. A proof of its correctness is included in an appendix. Using the shelling ordering concept, CAREL is shown to solve the exact probabilistic NCF measure for a hypercube in time polynomial in the number of spanning trees. However, this number increases exponentially in the hypercube dimension. This dissertation, then, aims to more efficiently obtain lower and upper bounds on the measures. Algorithms, presented in the text, generate tighter bounds than had been obtained previously and run in time polynomial in the cube dimension. The proposed algorithms for probabilistic 2CF measure consider PE and/or CE failures. In attempting to evaluate deterministic measures, a hybrid method for fault tolerant broadcasting in the hypercube is proposed. This method combines the favorable features of redundant and non-redundant techniques. A generalized result on the deterministic TBF measure for the hypercube is then described. Two distributed algorithms are proposed to identify the largest operational subcubes in a hypercube C\sb{n} with faulty PE\u27s. Method 1, called LOS1, requires a list of faulty components and utilizes the CMB operator of CAREL to solve the problem. In case the number of unavailable nodes (faulty or busy) increases, an alternative distributed approach, called LOS2, processes m available nodes in O(mn) time. The proposed techniques are simple and efficient

Efficient hypercube communications

Hypercube algorithms may be developed for a variety of communication-intensive tasks such as sending a message from one node to another, broadcasting a message from one node to all others, broadcasting a message from each node to all others, all-to-all personalized communication, one-to-all personalized communication, and exchanging messages between nodes via fixed permutations. All these communication patterns are special cases of many-to-many personalized communication. The problem of many-to-many personalized communication is investigated here. Two routing algorithms for many-to-many personalized communication are presented here. The algorithms proposed yield very high performance with respect to the number of time steps and packet transmissions. The first algorithm yields high performance through attempts to equibalance the number of messages at intermediate nodes. This technique tries to avoid creating a bottleneck at any node and thus reduces the total communication time. The second algorithm yields high performance through one-step time-lookahead equibalancing. It chooses from the candidate intermediate nodes the one which will probably have the minimum number of messages in the next cycle